Maximal estimate and integral operators in Bergman spaces with doubling measure

Abstract

The boundedness of the maximal operator on the upper half-plane Π + \Pi ^{+} is established. Here Π + \Pi ^+ is equipped with a positive Borel measure d ω ( y ) d x d\omega (y)dx satisfying the doubling property ω ( ( 0 , 2 t ) ) ≤ C ω ( ( 0 , t ) ) \omega ((0,2t))\leq C\omega ((0,t)) . This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces A ω p ( Π + ) A_{\omega }^{p}(\Pi ^{+}) .

Boundedness of area operators on anisotropic Hardy spaces

“…[1,4,10,24]). It is well known that the non-tangential area integral is a fundamental tool in complex analysis and harmonic analysis, which has many applications such as in the nontangential maximal functions, Littlewood-Paley theory, Volterra integral operators, and tent spaces (see, for instance, [9,10,[26][27][28]). The analysis of non-tangential area integral is developed from this.…”

Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

“…To give the precise statement of our main result, we need to introduce some notation. [13,14,16,18,[20][21][22]27].…”